Page:Elementary Text-book of Physics (Anthony, 1897).djvu/31

§ 17] point $$A$$. Draw the lines $$PQ$$ and $$PR$$ in the directions of the tangents at $$A$$ and $$B$$, equal to the velocities $$v_{0}$$ and $$v$$ of the point at $$A$$ and $$B$$ respectively. The line QB is the change in the velocity of the point during the time in which it traverses the distance $$AB$$. Draw the line $$QS$$ perpendicular to $$PR$$. The angle $$QPR$$, being the angle between the tangents at $$A$$ and $$B$$, equals the angle $$\alpha$$. In the limit, as $$\alpha$$ vanishes, $$v$$ and $$v_{0}$$ differ by the infinitesimal $$SR$$, and $$QS$$ equals $$v\alpha$$. The line $$SB$$ represents the change in the numerical magnitude of the velocity during the time $$t - t_{0}$$, and the rate of that change, which takes place along the tangent to the path, is given by The line $$QS$$ represents the change in velocity during the same time along the normal to the path. The acceleration along that normal is therefore $$\frac{v \alpha}{t - t_{0}}\cdot$$. Now under the conditions assumed in these statements $$AB = r \alpha$$, and $$\frac{AB}{t - t_{0}} = v$$, the velocity of the point. Hence $$v = \frac{r \alpha}{t - t_{0}}$$, and the acceleration along the normal to the path is If the path be a straight line, the normal acceleration vanishes, and the whole acceleration is given by the limit of the ratio $$\frac{v - v_{0}}{t - t_{0}} = \frac{dv}{dt}\cdot$$ If the path be a circle, and if the point move in it uniformly, the whole acceleration is given by $$\frac{v^2}{r}$$.

The unit of acceleration is that of a point, the velocity of which changes at a uniform rate by one unit of velocity in one second.

The dimensions of acceleration are $$LT^{-2}$$. Acceleration is completely described when its magnitude and